Resonator, unit having resonator, oscillator having unit and electronic apparatus having oscillator

ABSTRACT

A resonator comprises a base portion and a plurality of vibrational arms connected to the base portion and vibratable in a flexural mode. An electrode is disposed on at least one of opposite side surfaces of each of the vibrational arms so that a capacitance ratio r 1  of a fundamental mode of vibration of the resonator is less than a capacitance ratio r 2  of a second overtone mode of vibration thereof. A groove is formed in at least one of opposite main surfaces of each of the vibrational arms, and at least one mounting arm protrudes from the base portion.

FIELD OF THE INVENTION

The present invention relates to a resonator, a unit having the resonator, an oscillator having the unit and an electronic apparatus having the oscillator.

BACKGROUND OF THE INVENTION

There are many electronic apparatus comprising a display portion and a quartz crystal oscillator at least. For example, cellular phones, wristwatches, facsimiles, digital cameras and DVD recorders comprising a quartz crystal oscillator are well known. Recently, because of high stability for frequency, miniaturization and the light weight nature of these electronic apparatus, the need for an electronic apparatus comprising a smaller quartz crystal oscillator with a frequency of high stability has arisen. For example, the quartz crystal oscillator having a quartz crystal tuning fork resonator housed in a unit, which vibrates in a flexural mode, is widely used as a time standard in an electronic apparatus such as the cellular phones, the wristwatches, the facsimiles, the digital cameras and the DVD recorders.

Similar to this, the same need has also arisen for an electronic apparatus comprising a contour mode resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a SAW (Surface Acoustic Wave) resonator or a resonator for sensing angular velocity made of a piezoelectric material such as quartz crystal, lithium tantalite (LiTaO₃), lithium niobate (LiNbO₃) and ceramics.

Heretofore, however, it has been impossible to obtain an electronic apparatus comprising a smaller quartz crystal oscillator with a miniature quartz crystal tuning fork resonator of the prior art, capable of vibrating in a flexural mode, and having a frequency of high stability, a small series resistance and a high quality factor. This is the reason why, when miniaturized, the quartz crystal tuning fork resonator of the prior art, capable of vibrating in a flexural mode has a smaller electromechanical transformation efficiency. As a result, the resonator has a frequency of low stability, a large series resistance and a reduced quality factor.

Additionally, there has been a big problem in the quartz crystal oscillator of the prior art having the quartz crystal tuning fork resonator of the prior art, such that a frequency of a fundamental mode of vibration of the tuning fork resonator which is an output signal of the oscillator jumps to a second overtone mode of vibration thereof by shock or vibration.

Similarly, however, it has been impossible to obtain an electronic apparatus comprising a smaller quartz crystal oscillator with a contour mode resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a SAW resonator or a resonator for sensing angular velocity having a frequency of high stability, a small series resistance and a high quality factor because, when miniaturized, each resonator has a small electromechanical transformation efficiency, as a result, a frequency of low stability, a large series resistance and a low quality factor, and also is not strong against shock.

It is, therefore, a general object of the present invention to provide embodiments of a quartz crystal resonator, a quartz crystal unit, a quartz crystal oscillator and an electronic apparatus of the present invention, which overcome or at least mitigate one or more of the above problems.

SUMMARY OF THE INVENTION

The present invention relates to a resonator, a unit, an oscillator and an electronic apparatus comprising a display portion and a plurality of oscillators, one of which comprises a quartz crystal oscillator comprising a quartz crystal oscillating circuit having an amplification circuit and a feedback circuit, and in particular, relates to a quartz crystal resonator capable of vibrating in a flexural mode, a quartz crystal unit having the quartz crystal resonator and a quartz crystal oscillator having the quartz crystal unit and having an output signal of a frequency of high stability for a fundamental mode of vibration of the quartz crystal resonator, and also to a quartz crystal oscillator having a suppressed second overtone mode of vibration of the quartz crystal resonator, in addition, relates to a quartz crystal oscillator comprising an another contour mode resonator such as a length-extensional mode resonator, a width-extensional mode resonator and a Lame mode resonator or a thickness shear mode resonator, each made of quartz crystal or a SAW resonator or a piezoelectric resonator for sensing angular velocity. The quartz crystal oscillator is, therefore, available for the electronic apparatus requiring a miniature quartz crystal oscillator with high time accuracy and shock proof.

It is an object of the present invention to provide a miniature quartz crystal resonator, capable of vibrating in a flexural mode, and having a high electromechanical transformation efficiency.

It is an another object of the present invention to provide a miniature quartz crystal unit with a quartz crystal resonator, capable of vibrating in a fundamental mode of vibration of a flexural mode, and having a high electromechanical transformation efficiency.

It is a further object of the present invention to provide a quartz crystal oscillator with a miniature quartz crystal resonator, capable of vibrating in a flexural mode, and having a frequency of high stability, a small series resistance R₁ and a high quality factor Q₁, whose nominal frequency for a fundamental mode of vibration is within a range of 10 kHz to 200 kHz. Especially, a frequency of about 32.768 kHz is very available for a time standard of a frequency signal.

It is a still another object of the present invention to provide an electronic apparatus comprising a display portion and a plurality of oscillators.

According to one aspect of the present invention, there is provided a quartz crystal resonator comprising: a plurality of vibrational arms, each of the vibrational arms having a first main surface and a second main surface and side surfaces; and a base portion to which the vibrational arms are attached, in which the resonator has a piezoelectric constant e′₁₂ in the range of 0.1 C/m to 0.19 C/m in the absolute value.

According to a second aspect of the present invention, there is provided a quartz crystal unit comprising: a quartz crystal resonator having a base portion and a plurality of vibrational arms attached to the base portion; a case for housing the quartz crystal resonator; and a lid for covering an open end of the case, each of the vibrational arms having a first main surface and a second main surface opposite the first main surface and side surfaces, in which the quartz crystal resonator has a cutting angle in the range of ZY1wt(−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a third aspect of the present invention, there is provided a quartz crystal oscillator comprising: a quartz crystal oscillating circuit comprising; an amplification circuit comprising a CMOS inverter and a feedback resistor, and a feedback circuit comprising a quartz crystal resonator capable of vibrating in a flexural mode, a plurality of capacitors and a drain resistor, the quartz crystal resonator being housed in a package comprising a case for housing the quartz crystal resonator and a lid for covering an open end of the case, and comprising: a plurality of vibrational arms, each of the vibrational arms having a first main surface and a second main surface opposite the first main surface and side surfaces; and a base portion to which the vibrational arms are attached, in which the quartz crystal resonator has a cutting angle in the range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a fourth aspect of the present invention, there is provided an electronic apparatus comprising a display portion and a plurality of oscillators, one of the oscillators being a quartz crystal oscillator comprising: a quartz crystal oscillating circuit comprising; an amplification circuit having a CMOS inverter and a feedback resistor, and a feedback circuit having a quartz crystal tuning fork resonator capable of vibrating in a flexural mode of an inverse phase, a plurality of capacitors and a drain resistor, the quartz crystal tuning fork resonator comprising a tuning fork base and a plurality of tuning fork arms connected to the tuning fork base, each of the tuning fork arms having a first main surface and a second main surface opposite the first main surface and side surfaces, the quartz crystal tuning fork resonator being housed in a package comprising a case for housing the resonator and a lid for covering an open end of the case, in which the quartz crystal tuning fork resonator has a fundamental mode of vibration and a second overtone mode of vibration and the amplification circuit of the quartz crystal oscillating circuit has negative resistances −RL₁ and −RL₂ for the fundamental mode of vibration and the second overtone mode of vibration of the quartz crystal tuning fork resonator, in which an absolute value of the negative resistances is defined by |−RL₁| and |−RL₂| and a ratio of the |−RL₁| and R₁ is greater than that of the |−RL₂| and R₂, where R₁ and R₂ represent a series resistance of the fundamental mode of vibration and the second overtone mode of vibration of the quartz crystal resonator, respectively, in which an output signal of the quartz crystal oscillating circuit has an oscillation frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator and is a clock signal which is used to display time at a display portion of the electronic apparatus, and in which the quartz crystal tuning fork resonator has a cutting angle in the range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value.

Preferably, the piezoelectric constant e′₁₂ is within a range of 0.12 C/m² to 0.19 C/m² in the absolute value.

Preferably, mounting arms protruding from the base portion comprise two.

Preferably, each of the vibrational arms has a groove having a first stepped portion and a second stepped portion.

Preferably, each of the vibrational arms has a through groove.

Preferably, a merit value M₂ of a second overtone mode of vibration of the quartz crystal resonator having vibrational arms and a base portion is less than 30.

Preferably, the quartz crystal oscillator with the quartz crystal resonator is constructed so that a ratio of an amplification rate α₁ of the fundamental mode vibration and an amplification rate α₂ of the second overtone mode vibration of the amplification circuit is greater than that of a feedback rate β₂ of the second overtone mode vibration and a feedback rate β₁ of the fundamental mode vibration of the feedback circuit, and a product of the amplification rate α₁ and the feedback rate β₁ of the fundamental mode vibration is greater than 1.

The present invention will be more fully understood by referring to the following detailed specification and claims taken in connection with the appended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general view of a quartz crystal plate from which a quartz crystal resonator of the present invention is formed;

FIG. 2 shows a plan view of a quartz crystal resonator of a first embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 3 shows an A-A′ cross-sectional view of the vibrational arms of the quartz crystal resonator in FIG. 2;

FIG. 4 shows an A-A′ cross-sectional view of another embodiment of the vibrational arms of the quartz crystal resonator in FIG. 2;

FIG. 5 shows a plan view of a quartz crystal resonator of a second embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 6 shows a plan view of a quartz crystal resonator of a third embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 7 shows a B-B′ cross-sectional view of the vibrational arms of the resonator in FIG. 6;

FIG. 8 shows a plan view of a quartz crystal resonator of a fourth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 9 shows a plan view of a quartz crystal resonator of a fifth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 10 shows a plan view of a quartz crystal resonator of a sixth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 11 shows a plan view of a quartz crystal resonator of a seventh embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode;

FIG. 12 shows a plan view of a width-extensional mode quartz crystal resonator constructing an electronic apparatus of the present invention;

FIG. 13(a) and FIG. 13(b) show a plan view of a thickness shear mode quartz crystal resonator constructing an electronic apparatus of the present invention and a F-F′ sectional view of the resonator;

FIG. 14 shows a plan view of a Lame mode quartz crystal resonator constructing an electronic apparatus of the present invention;

FIG. 15(a) and FIG. 15(b) show a plan view of a resonator for sensing angular velocity constructing an electronic apparatus of the present invention and a G-G′ sectional view of the resonator;

FIG. 16(a) and FIG. 16(b) show a plan view of a quartz crystal resonator of an eighth embodiment of the present invention and comprising a quartz crystal tuning fork resonator, and a J-J′ sectional view of the resonator;

FIG. 17 shows a plan view of a quartz crystal unit of a first embodiment of the present invention and omitting a lid;

FIG. 18 shows a plan view of a quartz crystal unit of a second embodiment of the present invention and omitting a lid;

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a third embodiment of the present invention;

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of a first embodiment of the present invention;

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillating circuit constructing a quartz crystal oscillator of the present invention;

FIG. 22 shows a diagram of the feedback circuit of FIG. 21; and

FIG. 23 shows a block diagram of an embodiment of an electronic apparatus of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to the drawings, the embodiments of the present invention will be described in more detail.

FIG. 1 is a general view of a quartz crystal plate 1 from which a quartz crystal resonator of the present invention is formed, and particularly, a relationship of cutting angles θ_(x), θ_(y) and θ_(z) of the quartz crystal plate 1 and its coordinate system is illustrated in FIG. 1. The coordinate system has original point o, electrical axis x, mechanical axis y and optical axis z of quartz crystal and o-xyz is constructed.

First, a quartz crystal plate perpendicular to z axis, so called, Z plate quartz crystal is taken. The Z plate quartz crystal has a dimension of Width W₀, length L₀ and thickness T₀ corresponding to a respective direction of x, y and z axes.

Next, this Z plate quartz crystal is, first, rotated with an angle θ_(y) about the y axis, second, rotated with an angle θ_(x) about x′ axis which is a new axis of the x axis, and third, rotated with an angle θ_(z) about z″ axis which is a new axis of the z axis. In this case, each of the x, y and z axes changes to x″, y″ and z″ axes, respectively, because each axis is rotated twice about two axes. A quartz crystal resonator of the present invention is, therefore, formed from the quartz crystal plate with the rotation angles.

In other words, according to an expression of IEEE notation, a cutting angle of the quartz crystal resonator of the present invention can be expressed by ZY1wt(θ_(y))/(θ_(x))/(θ_(z)), and each of the angles θ_(y), θ_(x), θ_(z) will be described later in detail according to resonators of the present invention.

FIG. 2 shows a plan view of a quartz crystal resonator 10 of a first embodiment of the present invention and which is a quartz crystal tuning fork resonator. The resonator 10 comprises vibrational arms 20 and 31 and a base portion 40 attached to the vibrational arms, and the base portion 40 has mounting arms 36 and 37 protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms 20 and 31 has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arms 20 and 31 have grooves 21 and 27, respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator 10 has cutting angles θ_(y), θ_(x) and θ_(z) which are within a range of −20′ to +20°, −25° to +25° and −18° to +18°, respectively, namely, a cutting angle of the resonator is within a range of ZY1wt(−20′ to +20°)/(−25° to +25°)/(−18° to +18°). In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode of a fundamental mode of an inverse phase, and which is one of a contour mode quartz crystal resonator.

In more detail, the groove 21 is constructed to include a portion of a central linear line 41 of the arm 20, and the groove 27 is similarly constructed to include a portion of a central linear line 42 of the arm 31. Each of the grooves 21 and 27 has a width W₂, and the width W₂ including a portion of the central linear lines 41 and 42, is preferable because a large moment of inertia occurs at the arms 20 and 31 and the arms can vibrate in a flexural mode easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode can be obtained with a small series resistance R₁ and a high quality factor Q₁.

In addition, when each of the vibrational arms 20 and 31 has part widths W₁ and W₃, an arm width W of the arms 20 and 31 has a relationship of W=W₁+W₂+W₃, and the part widths W₁ and W₃ are constructed so that W₁≧W₃ or W₁<W₃. In addition, the width W₂ is constructed so that W₂≧W₁, W₃. In this embodiment, also, the grooves are constructed at the arms so that a ratio W₂/W of the width W₂ and the arm width W is greater than 0.35 and less than 1, preferably, within a range of 0.35 to 0.95 and a ratio t₁/t is less than 0.79, where t₁ and t are a thickness of the groove and the vibrational arms, as shown in FIG. 3, to obtain a very large moment of inertia of the vibrational arms. That is, the quartz crystal tuning fork resonator, capable of vibrating in the fundamental mode, and having a frequency of high stability can be provided with a small series resistance R₁, a high quality factor Q₁ and a small capacitance ratio r₁ because it has a very large electromechanical transformation efficiency.

Likewise, each of the vibrational arms 20 and 31 has a length L and each of the grooves 21 and 27 has a length lo (not shown here). In this embodiment, a ratio of the length I₀ and the length L is within a range of 0.3 to 0.8 to get a quartz crystal tuning fork resonator with series resistance R₁ of a fundamental mode of vibration smaller than series resistance R₂ of a second overtone mode of vibration. In other words, the length I₀ is within a range of 30% to 80% to the length L. In general, the length I₀ is within a range of 0.45 mm to 1.25 mm. Also, when a plurality of grooves are formed in at least one of upper and lower faces of the arms and divided into the length direction of the arms, the length I₀ is a total length of the grooves.

In addition, electrodes 25 and 26 are disposed on side surfaces of the vibrational arm 20 and an electrode 23 is disposed on a surface of the groove 21, which extends into the mounting arm 36 having a connecting portion 34. Similar to this, electrodes 32 and 33 are disposed on side surfaces of the vibrational arm 31 and an electrode 29 is disposed on a surface of the groove 27, which extends into the mounting arm 37 having a connecting portion 35. Each of the connecting portions 34 and 35 has a length L₂ and a width W_(S), and each of the mounting arms has a length L₃ and a width W₆. Also, the electrode 23 is connected to the electrodes 32 and 33, and the electrode 29 is connected to the electrodes 25 and 26.

In this embodiment, the length 10 of the groove corresponds to a length I_(d) of the electrode disposed inside each of the grooves, when the length I_(d) of the electrode is less than the length I₀ of the groove, namely, the length I₀ is of the length I_(d) of the electrode. In addition, the base portion 40 has a length L₁ and a width W_(H), the length L₁ is less than 0.5 mm, preferably, within a range of 0.015 mm to 0.49 mm, and a total length L_(t)(=L+L₁) in this embodiment is less than 2.1 mm, preferably, within a range of 1.02 mm to 1.95 mm to obtain a miniature quartz crystal resonator. Also, when a distance W₄ between the vibrational arms is taken, a total width W₅(=2W+W₄) is less than 0.53 mm, preferably, within a range of 0.15 mm to 0.52 mm, and the width W₅ is equal to or less than the W_(H) which is less than 0.55 mm, preferably, within a range of 0.15 mm to 0.53 mm.

In addition, the length L₃ is greater than or equal to the length L₂ and also, the length L₁ is greater than or equal to the length L₂ or the length L₁ is less than the length L₂. In actual, a value of L₁-L₂ is within a range of −0.1 mm to 0.32 mm, especially, when W_(H) is greater than W₅, a distance L₄ between an edge of the connecting portion and an outer edge of the vibrational arm is within a range of 0.012 mm to 0.38 mm. Also, the width W₆ is less than 0.45 mm and the length L₃ is less than 2.1 mm, preferably, within a range of 0.3 mm to 1.85 mm to reduce a leakage energy by vibration, and also, the width W₅ is less than 0.41 mm and the length L₂ is within a range of 0.04 mm to 0.5 mm to get a shock proof quartz crystal resonator having a reduced leakage energy by vibration.

Moreover, the distance W₄ and the width W₂ are constructed so that W₄≧W₂, and more, the distance W₄ is within a range of 0.045 mm to 0.65 mm and the width W₂ is within a range of 0.02 mm to 0.12 mm because it is easy to form a resonator shape and grooves formed at the vibrational arms separately by a photo-lithographic process and an etching process, consequently, a frequency stability for a fundamental mode of vibration of the resonator gets higher than that for a second overtone mode of vibration thereof. In this embodiment, a quartz wafer having a thickness t of 0.045 mm to 0.35 mm is used.

For example, in order to get a smaller-sized quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, it is necessary that the width W₂ of the groove is less than 0.07 mm and the arm width W is less than 0.18 mm, and preferably, the W is greater than 0.05 mm and less than 0.1 mm. Also, the thickness t₁ of the groove is within a range of 0.01 mm to 0.085 mm approximately, and the part widths W₁ and W₃ are less than 0.021 mm, respectively, preferably, less than 0.015 mm. In addition, a groove provided on at least one of an obverse face and a reverse face of the vibrational arms of this embodiment may be a through hole, namely, the thickness of the hole t₁=0.

In more detail, to obtain a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, and having a frequency of high stability which achieves high time accuracy, it is necessary to obtain the resonator whose resonance frequency is not influenced by shunt capacitance because quartz crystal is a piezoelectric material and the stability for frequency is very dependent on the shunt capacitance. In order to decrease the influence on the resonance frequency by the shunt capacitance, a merit value M_(i) plays an important role. Namely, the merit value M_(i) that expresses inductive characteristics, an electromechanical transformation efficiency and a quality factor of a quartz crystal tuning fork resonator, is defined by a ratio Q_(i)/r_(i) of a quality factor Q_(i) and capacitance ratio r_(i), namely, M_(i) is given by M_(i)=Q_(i)/r_(i), where i shows a vibration order of the resonator, and for example, when i=1 and 2, merit values M₁ and M₂ are for a fundamental mode of vibration of the resonator and a second overtone mode of vibration thereof, respectively.

Also, a frequency difference Δf of resonance frequency f_(s) of mechanical series independent on the shunt capacitance and resonance frequency f_(r) dependent on the shunt capacitance is inversely proportional to the merit value M_(i). The larger the value M_(i) becomes, the smaller the difference Δf becomes. Namely, the influence on the resonance frequency f_(r) by the shunt capacitance decreases because it is close to the resonance frequency f_(s). Accordingly, the larger the M_(i) becomes, the higher the stability for frequency of the quartz crystal tuning fork resonator becomes because the resonance frequency f_(r) of the resonator is almost never dependent on the shunt capacitance. Namely, the quartz crystal tuning fork resonator can be provided with a high time accuracy.

In detail, the quartz crystal tuning fork resonator can be obtained with the merit value M₁ of the fundamental mode of vibration greater than the merit value M₂ of the second overtone mode of vibration by providing the above-described tuning fork shape, grooves and dimensions. That is to say, a relationship of M₁>M₂ is obtained. As an example, when a resonance frequency of a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is about 32.768 kHz for a fundamental mode of vibration and the resonator has a value of W₂/W=0.5, t₁/t=0.34 and I₁/I=0.48, though there is a distribution in production, the resonator has a merit value of M₁>65 for the fundamental mode of vibration and a merit value of M₂<30 for the second overtone mode of vibration, respectively.

Namely, the quartz crystal tuning fork resonator can be provided with high inductive characteristics, good electromechanical transformation efficiency (small capacitance ratio r₁ and small series resistance R₁) and a high quality factor. As a result, a stability for frequency of the fundamental mode of vibration becomes higher than that of the second overtone mode of vibration, and simultaneously, the second overtone mode of vibration can be suppressed because capacitance ratio r₂ and series resistance R₂ of the second overtone mode of vibration become greater than capacitance ratio r₁ and series resistance R₁ of the fundamental mode of vibration, respectively. In particular, r₂ has a value greater than 1500 in this embodiment.

Therefore, the resonator capable of vibrating in the fundamental mode vibration can be provided with a high time accuracy because it has the frequency of high stability. Consequently, a quartz crystal oscillator comprising the quartz crystal tuning fork resonator of this embodiment outputs an oscillation frequency of the fundamental mode vibration as an output signal, and the frequency of the output signal has a very high stability, namely, excellent time accuracy. In other words, the quartz crystal oscillator of this embodiment has a remarkable effect such that a frequency change by ageing becomes extremely small. Also, an oscillation frequency of the quartz crystal resonator of this embodiment is adjusted so that a frequency deviation is within a range of −100 PPM to +100 PPM to a nominal frequency, e.g. 32.768 kHz, after mounting it on a mounting portion of a case for housing the resonator.

In addition, the groove thickness t₁, shown in FIG. 3, of the present invention is the thinnest thickness of the grooves because quartz crystal is an anisotropic material and the groove thickness t₁ has a distribution when it is formed by a chemical etching method. In detail, a groove shape of the sectional view of vibrational arms in FIG. 3 has a rectangular shape, but the groove shape has an about U shape actually. In the above-described embodiments, though the grooves are constructed at the arms, this invention is not limited to this, namely, a relationship of the merit values M₁ and M₂ can be applied to the conventional quartz crystal tuning fork resonator and a relationship of a quartz crystal oscillating circuit comprising an amplification circuit and a feedback circuit can be also applied to the conventional quartz crystal tuning fork resonator to suppress a second overtone mode vibration and to get a high frequency stability for a fundamental mode of vibration of the quartz crystal tuning fork resonator.

FIG. 3 shows an A-A′ cross-sectional view of the vibrational arms 20 and 31 of the quartz crystal resonator 10 in FIG. 2, and electrode construction within the grooves. The vibrational arm 20 has grooves 21 and 22 cut into it, which include a portion of central linear line of the arm 20. The grooves 21 and 22 have a first set of electrodes 23 and 24 of the same electrical polarity, while the side surfaces of the arm 20 have a second set of electrodes 25 and 26 having an opposite electrical polarity to the first set of electrodes 23 and 24. The vibrational arm 31 has grooves 27 and 28 constructed in a similar manner as the vibrational arm 20. The grooves 27 and 28 have a third set of electrodes 29 and 30 of the same electrical polarity, and the side surfaces of the vibrational arm 31 have a fourth set of electrodes 32 and 33 with the opposite electrical polarity to the third electrodes 29 and 30. The electrodes disposed on the vibrational arms 20 and 31 are connected as shown in FIG. 3, namely, two electrode terminals of different electrical polarity C-C′ are obtained.

In detail, the first set of electrodes 23 and 24 disposed on the grooves 21 and 22 of the vibrational arm 20 have the same electrical polarity as the fourth set of electrodes 32 and 33 disposed on both side surfaces of the vibrational arm 31, while the second set of electrodes 25 and 26 disposed on both side surfaces of the vibrational arm 20 have the same electrical polarity as the third set of electrodes 29 and 30 disposed on the grooves 27 and 28 of the arm 31. When a direct voltage is applied between the electrode terminals C-C′, an electric field Ex occurs along the arrow direction inside the vibrational arms 20 and 31. As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R₁ and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator.

FIG. 4 shows an A-A′ cross-sectional view of another embodiment of the vibrational arms 20 and 31 of the quartz crystal resonator 10 in FIG. 2. The vibrational arm 20 has a through groove 21 a, which include a portion of central linear line of the arm 20. The through groove 21 a has a first set of electrodes 23 a and 24 a of the same electrical polarity, while the side surfaces of the arm 20 have a second set of electrodes 25 and 26 having an opposite electrical polarity to the first set of electrodes 23 a and 24 a. The vibrational arm 31 has a through groove 27 a constructed in a similar manner as the vibrational arm 20. The through groove 27 a has a third set of electrodes 29 a and 30 a of the same electrical polarity, and the side surfaces of the vibrational arm 31 have a fourth set of electrodes 32 and 33 with the opposite electrical polarity to the third electrodes 29 a and 30a. The electrodes disposed on the vibrational arms 20 and 31 are connected as shown in FIG. 4, namely, two electrode terminals of different electrical polarity E-E′ are obtained.

In detail, the first set of electrodes 23 a and 24 a disposed on the through groove 21 a of the vibrational arm 20 have the same electrical polarity as the fourth set of electrodes 32 and 33 disposed on both side surfaces of the vibrational arm 31, while the second set of electrodes 25 and 26 disposed on both side surfaces of the vibrational arm 20 have the same electrical polarity as the third set of electrodes 29 a and 30 a disposed on the through groove 27 a of the arm 31. When a direct voltage is applied between the electrode terminals E-E′, an electric field Ex occurs along the arrow direction inside the vibrational arms 20 and 31. As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R₁ and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator.

FIG. 5 shows a plan view of a quartz crystal resonator 50 of a second embodiment of the present invention, and which is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. The resonator 50 comprises vibrational arms 60 and 71 and a base portion 80 attached to the vibrational arms, and the base portion 80 has a mounting arm 77 protruding from the base portion, and the mounting arm 77 is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms 60 and 71 has a first main surface and a second main surface and side surfaces, and the vibrational arms 60 and 71 have grooves 61 and 67, respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator 50 has the same cutting angles θ_(y), θ_(x) and θ_(z) and the same dimensions W₁, W₂, W₃, W₄, W₅, W, L₁, L₂, L₃, L₄ and L as the resonator of FIG. 2.

In more detail, the groove 61 is constructed to include a portion of a central linear line 81 of the arm 60, and the groove 67 is similarly constructed to include a portion of a central linear line 82 of the arm 71. Each of the grooves 61 and 67 has a width W₂, and the width W₂ includes a portion of the central linear lines 81 and 82 because a large moment of inertia occurs at the arms 60 and 71. In this embodiment, the resonator is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode and which can vibrate in a fundamental mode of an inverse phase easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode of an inverse phase can be obtained with a small series resistance R₁ and a high quality factor Q₁.

In addition, electrodes 65 and 66 are disposed on side surfaces of the vibrational arm 60 and an electrode 63 is disposed on a surface of the groove 61, and which is connected to electrodes 72 and 73 disposed on side surfaces of the vibrational arm 71. Similar to this, an electrode 69 is disposed on a surface of the groove 67, which extends into the mounting arm 77 having a connecting portion 75. Also, the electrode 69 is connected to the electrodes 65 and 66.

FIG. 6 shows a plan view of a quartz crystal resonator 90 of a third embodiment of the present invention, which is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. The resonator 90 comprises vibrational arms 91 and 92 and a base portion 95 attached to the vibrational arms, and the base portion 90 has a mounting arm 96 protruding from the base portion. In this embodiment, the mounting arm 96 is between the vibrational arms 91 and 92 and is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms 91 and 92 has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arms 91 and 92 have grooves 93 and 94, respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator 90 has the same cutting angles θ_(y), θ_(x) and θ_(z) as the quartz crystal resonator 10 of FIG. 2. In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode of a fundamental mode of an inverse phase.

In detail, similar to the resonator of FIG. 2, the groove 93 is also constructed to include a portion of a central linear line of the arm 91, and also, the groove 94 is similarly constructed to include a portion of a central linear line of the arm 92. Each of the grooves 93 and 94 has a width W₂, and the width W₂ includes a portion of the central linear lines because a large moment of inertia occurs at the vibrational arms 91 and 92 and which can vibrate in a flexural mode easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode can be obtained with a small series resistance R₁ and a high quality factor Q₁.

In addition, the base portion 95 has a length L₁ and a width W_(H). Also, each of the vibrational arms 91 and 92 has a width W, part widths W₁ and W₃ and a width W₂ of the groove, namely, there is a relationship of W=W₁+W₂+W₃. In detail, the resonator 90 has the same dimensions L₁, W_(H), W₁, W₂, W₃, and W and the same relationship as the resonator 10 of FIG. 2. Moreover, the mounting arm 96 has a width W₁₁ and there is a distance W₁₀ between the vibrational arm 91 or the vibrational arm 92 and the mounting arm 96. In order to get a quartz crystal tuning fork resonator with reduced leakage energy by vibration of the vibrational arms, the distance W₁₀ is within a range of 0.032 mm to 0.21 mm and the width W₁ is within a range of 0.21 mm to 0.88 mm. In addition, to get a shockproof quartz crystal tuning fork resonator, W₁₁ has a relationship of (1.2 to 7.6)×W. In this embodiment, a total width W_(t)(=2 W+2 W₁₀+W₁₁) is less than 1.3 mm, preferably, within a range of 0.52 mm to 1.2 mm, to get a miniature quartz crystal tuning fork resonator.

FIG. 7 shows a B-B′ cross-sectional view of the vibrational arms 91 and 92 of the resonator 90 in FIG. 6. The vibrational arm 91 has grooves 93 and 97 cut into it, and the grooves 93 and 97 have a first set of electrodes 100 and 101 of the same electrical polarity, while the side surfaces of the arm 91 have a second set of electrodes 99 and 102 having an opposite electrical polarity to the first set of electrodes 100 and 101. The vibrational arm 92 has grooves 94 and 98 constructed in a similar manner as the vibrational arm 91. The grooves 94 and 98 have a third set of electrodes 106 and 107 of the same electrical polarity, and the side surfaces of the vibrational arm 92 have a fourth set of electrodes 105 and 108 with the opposite electrical polarity to the third electrodes 106 and 107. The electrodes disposed on the vibrational arms 91 and 92 are connected as shown in FIG. 7, namely, two electrode terminals of different electrical polarity D-D′ are obtained.

In detail, the first set of electrodes 100 and 101 disposed on the grooves 93 and 97 of the vibrational arm 91 have the same electrical polarity as the fourth set of electrodes 105 and 108 disposed on both side surfaces of the vibrational arm 92 and an electrode 104 disposed on the mounting arm 96, while the second set of electrodes 99 and 102 disposed on both side surfaces of the vibrational arm 91 have the same electrical polarity as the third set of electrodes 106 and 107 disposed on the grooves 94 and 98 of the arm 92 and an electrode 103 disposed on the mounting arm 96. When a direct voltage is applied between the electrode terminals D-D′, an electric field Ex occurs along the arrow direction inside the vibrational arms 91 and 92. As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. Consequently, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R₁ and a high quality factor Q because even when miniaturized, there is very large electromechanical transformation efficiency for the resonator.

FIG. 8 shows a plan view of a quartz crystal resonator 110 of a fourth embodiment of the present invention, which is a quartz crystal tuning fork resonator. The resonator 110 comprises vibrational arms 111 and 112 and a base portion 113 attached to the vibrational arms, and the base portion 113 has mounting arms 114 and 115 protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, the vibrational arms 111 and 112 have grooves 116 and 117, respectively. In more detail, each of the vibrational arms 111 and 112 has an end portion connected to the base portion and a free end portion, when a distance measured from the end portion to the free end portion is a length L, each of the vibrational arms has a width W between the end portion and half a length L/2 of each of the vibrational arms and a width W_(e) between half the length L/2 and the length L of the free end portion of each of the vibrational arms, and the width W is less than the width W_(e). In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode and vibrate in a fundamental mode of an inverse phase.

Similar to this, a quartz crystal tuning fork resonator 120 of a fifth embodiment of the present invention is shown in FIG. 9 showing a plan view thereof. The resonator 120 comprises vibrational arms 121 and 122 and a base portion 123 attached to the vibrational arms, and the base portion 123 has a mounting arm 124 protruding from the base portion. In this embodiment, the mounting arm 124 is between the vibrational arms 121 and 122 and the vibrational arms 121 and 122 have grooves 125 and 126, respectively. For this case the width W is also less than the width W_(e) similar to that of FIG. 8.

FIG. 10 shows a plan view of a quartz crystal resonator 130 of a sixth embodiment of the present invention, which is a quartz crystal tuning fork resonator. The resonator 130 comprises vibrational arms 131 and 132 and a base portion 133 attached to the vibrational arms, and the base portion 133 has mounting arms 134 and 135 protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, the vibrational arms 131 and 132 have grooves 136 and 137, respectively. In more detail, each of the vibrational arms 131 and 132 has an end portion connected to the base portion and a free end portion, when a distance measured from the end portion to the free end portion is a length L, each of the vibrational arms has a width W between the end portion and half a length L/2 of each of the vibrational arms and a width W_(e) between half the length L/2 and the length L of the free end portion of each of the vibrational arms, and the width W is less than the width W_(e). In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode and vibrate in a fundamental mode of an inverse phase.

Similar to this, a quartz crystal tuning fork resonator 140 of a seventh embodiment of the present invention is shown in FIG. 11 showing a plan view thereof. The resonator 140 comprises vibrational arms 141 and 142 and a base portion 143 attached to the vibrational arms, and the base portion 143 has a mounting arm 144 protruding from the base portion. In this embodiment, the mounting arm 144 is between the vibrational arms 141 and 142 and the vibrational arms 141 and 142 have grooves 145 and 146, respectively. For this case the width W is also less than the width W_(e) similar to that of FIG. 10.

Especially, the quartz crystal tuning fork resonator of the fourth embodiment to the seventh embodiment can be miniaturized with a small series resistance R₁ and a high quality factor Q₁ because it has a width W_(e) greater than a width W, and the width W_(e) operates as a mass.

Next, a value of a piezoelectric constant e′₁₂ is described, which is of great importance and necessary to excite a quartz crystal tuning fork resonator capable of vibrating in a flexural mode of the present invention. The larger a value of the piezoelectric constant e′₁₂ becomes, the higher electromechanical transformation efficiency becomes. The piezoelectric constant e′₁₂ of the present invention can be defined by a function of the cutting angles θ_(y), θ_(x) and θ_(z) shown in FIG. 1, and piezoelectric constants e₁₁=0.171 C/m² and e₁₄=−0.0406 C/m² of quartz crystal. In order to obtain a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode and having a small series resistance R₁ and a high quality factor Q, the piezoelectric constant e′₁₂ of the present invention is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value. It is, therefore, easily understood that this value is enough large to obtain the quartz crystal tuning fork resonator with a small series resistance R₁ and a high quality factor Q.

Especially, in order to obtain a quartz crystal tuning fork resonator capable of vibrating in a flexural mode with a much smaller series resistance R₁, the piezoelectric constant e′₁₂ is preferably within a range of 0.12 C/m² to 0.19 C/m² in the absolute value.

In addition, as an example, a quartz crystal tuning fork resonator comprises a plurality of vibrational arms having a first vibrational arm and a second vibrational arm, and a groove having a first stepped portion and a second stepped portion is formed in at least one of a first main surface and a second main surface of each of the first and second vibrational arms, in which a first electrode is disposed on the first stepped portion of the groove, a second electrode is disposed on the second stepped portion of the groove, and a third electrode is disposed on each of side surfaces of each of the first and second vibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) is between the first electrode and the third electrode disposed opposite to the first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) is between the second electrode and the third electrode disposed opposite to the second electrode, and in which the piezoelectric constants e′_(12i) and e′_(12o) are within the range of 0.12 C/m² to 0.19 C/m² in the absolute value, respectively, and a product of the e′_(12i) and the e′_(12o) is greater than 0.

As an another example, a quartz crystal tuning fork resonator comprises a plurality of vibrational arms having a first vibrational arm and a second vibrational arm, and a through hole having a first side surface and a second side surface is formed in each of the first and second vibrational arms, in which a first electrode is disposed on the first side surface of the through hole, a second electrode is disposed on the second side surface of the through hole, and a third electrode is disposed on each of side surfaces of each of the first and second vibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) is between the first electrode and the third electrode disposed opposite to the first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) is between the second electrode and the third electrode disposed opposite to the second electrode, and in which the piezoelectric constants e_(12i) and e_(12o) are within the range of 0.12 C/m² to 0.19 C/m² in the absolute value, respectively, and a product of the e′_(12i) and the e′_(12o) is greater than 0.

Therefore, the quartz crystal tuning fork resonator described above has a small series resistance R₁ and a high quality factor Q, and also a frequency of high stability.

FIG. 12 shows a plan view of a width-extensional mode quartz crystal resonator 150 constructing an electronic apparatus of the present invention, and which vibrates in a width-extensional mode. The width-extensional mode resonator 150 comprises a vibrational portion 151, connecting portions 152, 152 a and supporting portions 153 and 154 connected to a mounting portion 155 constructing the supporting portions. Namely, the vibrational portion 151 is connected to the supporting portions 153, 154 having the mounting portion 155 through the connecting portions 152, 152 a. In addition, an electrode 151 a is disposed on an obverse surface of the vibrational portion 151 and an electrode 151 b (not visible) is disposed on a reverse surface of the vibrational portion 151.

In more detail, the electrode 151 a disposed on the obverse surface of the vibrational portion 151 is connected to an electrode 153 a disposed on the mounting portion 155, while the electrode 151 b disposed on the reverse surface of the vibrational portion 151 is connected to an electrode 154 a disposed on the mounting portion 155 through an electrode 154 b disposed on a side surface of the mounting portion. Namely, a pair of electrodes is disposed on the obverse and reverse surfaces of the vibrational portion 151. Also, the vibrational portion 151 has a width W₀ and a length L₀. In general, a ratio W₀/L₀ is less than 0.35. In addition, the mounting portion 155 is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (80° to 100°)/(−10° to +10°)/(75° to +115°) and a piezoelectric constant e′₃₁ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value to obtain the resonator with a small series resistance R₁ and a high quality factor Q.

Similar to this, a length-extensional mode quartz crystal resonator can be obtained by replacing the width W₀ with the length L₀. In this case the connecting portions are formed in the width direction. In this embodiment, a cutting angle of the length-extensional mode resonator is within a range of ZY1wt (80° to 100°)/(−10° to +10°)/(−35° to +35°) and a piezoelectric constant e′₃₂ of the resonator is within a range of 0.12 C/m² to 0.19 C/m² in the absolute value to obtain the resonator with a small series resistance R₁ and a high quality factor Q.

FIG. 13(a) and FIG. 13(b) show a plan view of a thickness shear mode quartz crystal resonator 156 constructing an electronic apparatus of the present invention and a F-F′ sectional view of the thickness shear mode resonator 156 capable of vibrating in a thickness shear mode. The resonator 156 comprises a vibrational portion 157 having a width W₀, a length L₀ and a thickness T₀, and electrodes 158 and 159 are disposed on an obverse surface and a reverse surface so that the electrodes have opposite electrical polarity each other. Namely, a pair of electrodes is disposed on the vibrational portion 157. Also, the resonator 156 is housed in a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (−5° to +5°)/±(37° to 58°)/(85° to 95°) and a piezoelectric constant e′₃₄ of the resonator is within a range of 0.055 C/m² to 0.14 C/m² in the absolute value to obtain the resonator with a small series resistance R₁ and a high quality factor Q. In order to obtain much smaller series resistance R₁, the piezoelectric constant e′₃₄ of the resonator is preferably within a range of 0.085 C/m² to 0.12 C/m² in the absolute value.

FIG. 14 shows a plan view of a Lame mode quartz crystal resonator 210 constructing an electronic apparatus of the present invention, and vibrating in a Lame mode. The Lame mode resonator 210 comprises a vibrational portion 211, connecting portions 212, 213 and supporting portions 214, 215 having mounting portions 216 and 217. Namely, the vibrational portion 211 is connected to the supporting portions 214, 215 through the connecting portions 212, 213. In addition, an electrode 218 is disposed on an obverse surface of the vibrational portion 211 and an electrode 219 (not visible) is disposed on a reverse surface of the vibrational portion 211.

In more detail, the electrode 218 disposed on the obverse surface of the vibrational portion 211 is extended into the mounting portion 217, while the electrode 219 disposed on the reverse face of the vibrational portion 211 is extended into the mounting portion 216. Namely, a pair of electrodes is disposed on the obverse and reverse surfaces of the vibrational portion 151. Also, the vibrational portion 211 has a width W₀ and a length L₀. In general, a ratio L₀/W₀ is approximately equal to m for a fundamental mode of vibration and an overtone mode of vibration of the resonator, where m is an order of vibration of the resonator and an integer. For example, when a Lame mode quartz crystal resonator has one of m=1, 2, 3 and n, the resonator vibrates in a fundamental mode for m=1, a second overtone mode for m=2, a third overtone mode for m=3 and an n^(th) overtone mode for m=n.

Also, the m has a close relationship to the number of electrodes disposed on the vibrational portion. For example, when an electrode is disposed opposite each other on both of an obverse surface and a reverse surface of the vibrational portion, this is called “the number of two electrodes”, in other words, “a pair of electrodes”. Namely, the vibrational portion has the number of p electrodes, where p is an even number such as 2, 4, 6, 8 and 10. As an example, when p of the vibrational portion comprises 6, the vibrational portion vibrates in a third overtone mode. In this example, three pairs of electrodes are disposed on the vibrational portion. Namely, the third overtone mode of vibration is a principal vibration. As is apparent from this relationship, there is a relationship of m=p/2.

Therefore, the principal vibration vibrates in the order of vibration corresponding to the number of an electrode pair or electrode pairs. For example, the principal vibration vibrates in a fundamental mode of vibration, a second overtone mode of vibration and a third overtone mode of vibration, respectively, for an electrode pair, two electrode pairs and three electrode pair disposed on the vibrational portion. In detail, when me electrode pairs are disposed on the vibrational portion, the principal vibration vibrates in an n^(th) overtone mode of vibration, and m_(e) corresponds to the n, where m_(e) is an integer. It is needless to say that this concept can be applied to a width-extensional mode quartz crystal resonator and a length-extensional mode quartz crystal resonator.

In more detail, an even number of electrodes are disposed on the obverse and reverse surfaces of the vibrational portion and the electrodes disposed opposite each other has an opposite electrical polarity. In addition, each of the mounting portions 216, 217 is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (−5° to +5°)/±(35° to 60°)/±(40° to 50°) and a piezoelectric constant e′₃₂ of the resonator is within a range of 0.045 C/m² to 0.13 C/m² in the absolute value to obtain the resonator with a small series resistance R₁ and a high quality factor Q.

FIG. 15(a) and FIG. 15(b) show a plan view of a resonator for sensing angular velocity constructing an electronic apparatus of the present invention and a G-G′ sectional view of the resonator. In this embodiment, the resonator 220 comprises a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode and comprising vibrational arms 221, 222 and a base portion 223 attached to the vibrational arms, the base portion 223 is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms 221 and 222 has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arm 221 has grooves 224, 230, while the vibrational arm 222 has grooves 225, 237, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, a cutting angle of the resonator is within a range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18) and a piezoelectric constant e′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in the absolute value. The resonator of this embodiment can vibrate in a flexural mode of a fundamental mode and an inverse phase.

In more detail, electrodes 226 and 227 which are of the same electrical polarity are disposed on the side surfaces such that an electrode terminal H is defined, while an electrode 228 is disposed inside the groove 224 and an electrode 229 which is of the same electrical polarity to the electrode 228 is disposed inside the groove 230 such that an electrode terminal H′ of opposite electrical polarity to the electrode terminal H is defined. Namely, two electrode terminals H-H′ for an input signal are constructed. On the other hand, electrodes 231, 232, 233 which are of the same electrical polarity are disposed on the side surfaces and inside the grooves 225, 237 such that an electrode terminal I is defined, while electrodes 234, 235, 236 which are of the same electrical polarity are disposed on the side surfaces and inside the grooves 225, 237 such that an electrode terminal I′ of opposite electrical polarity to the electrode terminal I is defined. Namely, two electrode terminals I-I′ for an output signal are constructed. The resonator of this embodiment is made of quartz crystal, but may be made of a piezoelectric material such as lithium tantalite, lithium niobate and ceramics.

FIG. 16(a) and FIG. 16(b) show a plan view and a J-J′ sectional view of a quartz crystal resonator 230 of an eighth embodiment of the present invention, and which is a quartz crystal tuning fork resonator. The resonator 230 comprises vibrational arms 231 and 232 and a base portion 233 attached to the vibrational arms. In addition, each of the vibrational arms 231 and 232 has a first main surface and a second main surface opposite the first main surface and side surfaces, and the first and second main surfaces have central linear portions 242, 243, respectively. The vibrational arm 231 has grooves 234, 235 and the vibrational arm 232 has grooves 235, 237. In addition, the grooves 234 and 236 are formed between the central linear portion 242 and an outer edge of the vibrational arm 231, respectively. Similar to this, the grooves 235 and 237 are formed between the central linear portion 243 and an outer edge of the vibrational arm 232, respectively

Moreover, each of the grooves 234, 235, 236 and 237 has a width W₈, and a width W₇ including a portion of the central linear lines 242 and 243 is formed in each of the vibrational arms 231 and 232. In addition, a distance W₉ is formed in the width direction of the vibrational arms 231 and 232 between an outer edge of the groove and an outer edge of the vibrational arms. In detail, a width W of the arms 231 and 232 has generally a relationship of W=W₇+2W₈+2W₉, and the width W₈ is constructed so that W₈≧W₇, W₉. In this embodiment, also, the grooves are constructed at the arms so that a ratio W₈/(W/2) of the width W₈ and a half width of the width W is greater than 0.35 and less than 1, preferably, within a range of 0.35 to 0.95. In addition, the width W₇ is less than 0.05 mm, preferably, less than 0.03 mm and the width W₈ is within a range of 0.015 mm to 0.05 mm to obtain a very large moment of inertia of the vibrational arms. That is, the quartz crystal tuning fork resonator, capable of vibrating in a fundamental mode, and having a frequency of high stability can be provided with a small series resistance R₁, a high quality factor Q₁ and a small capacitance ratio r₁ because it has a very large electromechanical transformation efficiency.

In FIG. 16(b), the vibrational arm 231 has grooves 234, 236, 238 and 240 cut into it. The grooves 234, 236, 238 and 240 have a first set of electrodes 256, 257, 258 and 259 of the same electrical polarity, while the side surfaces of the arm 231 have a second set of electrodes 244 and 245 having an opposite electrical polarity to the first set of electrodes 256, 257, 258 and 259. The vibrational arm 232 has grooves 235, 237, 239 and 241 constructed in a similar manner as the vibrational arm 231. The grooves 235, 237, 239 and 241 have a third set of electrodes 252, 253, 254 and 255 of the same electrical polarity, and the side surfaces of the vibrational arm 232 have a fourth set of electrodes 250 and 251 with the opposite electrical polarity to the third electrodes 252, 253, 254 and 255. The electrodes disposed on the vibrational arms 231 and 232 are connected as shown in FIG. 16(b), namely, two electrode terminals of different electrical polarity K-K′ are obtained.

In detail, the first set of electrodes 256, 257, 258 and 259 disposed on the grooves 234, 236, 238 and 240 of the vibrational arm 231 have the same electrical polarity as the fourth set of electrodes 250 and 251 disposed on both side surfaces of the vibrational arm 232, while the second set of electrodes 244 and 245 disposed on both side surfaces of the vibrational arm 231 have the same electrical polarity as the third set of electrodes 252, 253, 254 and 255 disposed on the grooves 235, 237, 239 and 241 of the arm 232. When a direct current voltage is applied between the electrode terminals K-K′, an electric field Ex occurs along the arrow direction inside the vibrational arms 231 and 232. As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R₁ and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator.

FIG. 17 shows a plan view of a quartz crystal unit of a first embodiment of the present invention and omitting a lid. The quartz crystal unit 160 comprises the quartz crystal tuning fork resonator 10 shown in FIG. 2, a case 165 for housing the resonator and a lid for covering an open end of the case (not shown here). Also, the resonator 10 has mounting arms 36 and 37, each of which is mounted on a mounting portion 166 and a mounting portion 167 of the case 165. In detail, an electrode disposed on the mounting arm 36 is connected to an electrode disposed on the mounting portion 166 by adhesives 168 or a metal such as solder, and similarly, an electrode disposed on the mounting arm 37 is connected to an electrode disposed on the mounting portion 167 by adhesives 169 or a metal.

FIG. 18 shows a plan view of a quartz crystal unit of a second embodiment of the present invention and omitting a lid. The quartz crystal unit 170 comprises the quartz crystal tuning fork resonator 50 shown in FIG. 5, a case 175 for housing the resonator and a lid for covering an open end of the case (not shown here). Namely, the resonator comprises vibrational arms and a base portion. Also, the case 175 has mounting portions 176 and 177 and the resonator 50 has a mounting arm 77 protruding from the base portion, which is mounted on the mounting portion 177 of the case 175. In detail, an electrode disposed on the mounting arm 77 is connected to an electrode disposed on the mounting portion 177 by adhesives 179 or a metal such as solder, and similarly, an electrode disposed on the base portion of the resonator 50 is connected to an electrode disposed on the mounting portion 176 by adhesives 178 or a metal such as solder.

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a third embodiment of the present invention. The quartz crystal unit 180 comprises a contour mode quartz crystal resonator 185 or a thickness shear mode quartz crystal resonator 185, a case 181 and a lid 182. In more detail, the resonator 185 is mounted on a mounting portion 184 of the case 181 by conductive adhesives 76 or solder. Also, the case 181 and the lid 182 are connected through a connecting member 183. For example, the contour mode resonator 185 in this embodiment is the same resonator as one of the quartz crystal tuning fork resonators 10, 50, 90, 110, 120, 130, 140, 220 and 230 described in detail in FIG. 2-FIG. 11 and FIG. 15-FIG. 16.

In this embodiment, circuit elements are connected at outside of the quartz crystal unit to get a quartz crystal oscillator. Namely, only the quartz crystal tuning fork resonator is housed in the unit and also, it is housed in the unit in vacuum. In this embodiment, the quartz crystal unit of a surface mounting type is shown, but the quartz crystal tuning fork resonator may be housed in a tubular type, namely, a quartz crystal unit of the tubular type. In addition, the quartz crystal unit of the present invention includes any shape of a quartz crystal unit comprising a quartz crystal resonator, a case and a lid to house the quartz crystal resonator in vacuum.

Also, instead of the quartz crystal tuning fork resonator and the thickness shear mode quartz crystal resonator, one of a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator which are a contour mode quartz crystal resonator, respectively, or a SAW (Surface Acoustic Wave) resonator or a piezoelectric resonator for sensing angular velocity (angular velocity sensor) made of quartz crystal or ceramics may be housed in the unit.

In addition, a member of the case and the lid is ceramics or glass and a metal or glass, respectively, and a connecting member is a metal or glass with low melting point. Also, a relationship of the resonator, the case and the lid described in this embodiment is applied to a quartz crystal oscillator of the present invention which will be described in FIG. 20.

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of a first embodiment of the present invention. The quartz crystal oscillator 190 comprises a quartz crystal oscillating circuit, a case 191 and a lid 192. In this embodiment, circuit elements constructing the oscillating circuit are housed in a quartz crystal unit comprising a contour mode quartz crystal resonator 195 or a thickness shear mode quartz crystal resonator 195, the case 191 and the lid 192. Also, the oscillating circuit of this embodiment comprises an amplifier 197 including a feedback resistor, the resonator 195, a plurality of capacitors (not shown here) and a drain resistor (not shown here), and a CMOS inverter is used as the amplifier 197.

In addition, in this embodiment, the resonator 195 is mounted on a mounting portion 194 of the case 191 by conductive adhesives 196 or solder. As described above, the amplifier 197 is housed in the quartz crystal unit and mounted on the case 191. Also, the case 191 and the lid 192 are connected through a connecting member 193.

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillating circuit constructing a quartz crystal oscillator of the present invention. The quartz crystal oscillating circuit 201 comprises an amplifier (CMOS inverter) 202, a feedback resistor 204, a drain resistor 207, a plurality of capacitors 205, 206 and a quartz crystal resonator 203. Namely, the oscillating circuit 201 comprises an amplification circuit 208 having the amplifier 202 and the feedback resistor 204, and a feedback circuit 209 having the drain resistor 207, the capacitors 205, 206 and the quartz crystal resonator 203. The quartz crystal resonator 203 is one of the resonators already described above. For example, when the resonator 203 is a quartz crystal tuning fork resonator capable of vibrating in a flexural mod, an output signal of the oscillating circuit 201 is outputted through a buffer circuit (not shown in FIG. 21), and is an oscillating frequency of a fundamental mode of vibration of the resonator.

In other words, the oscillation frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator is outputted from the oscillating circuit through the buffer circuit as an output signal. According to the present invention, a nominal frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator is within a range of 10 kHz to 200 kHz. Especially, a frequency of 32.768 kHz is very available for use in an electronic apparatus of the present invention. In general, the output signal has an oscillation frequency which is within a range of −100 PPM to +100 PPM to the nominal frequency, e.g. 32.768 kHz.

In more detail, the quartz crystal oscillator in this example comprises a quartz crystal oscillating circuit and a buffer circuit, namely, the quartz crystal oscillating circuit comprises an amplification circuit and a feedback circuit, and the amplification circuit comprises an amplifier (CMOS inverter) and a feedback resistor and the feedback circuit comprises a quartz crystal tuning fork resonator capable of vibrating in a flexural mode, a drain resistor and a plurality of capacitors. Also, the quartz crystal tuning fork resonator already described in FIG. 2-FIG. 11 and FIG. 15-FIG. 16 is used in a quartz crystal oscillator of the present invention. Instead of the quartz crystal tuning fork resonator, an another contour mode quartz crystal resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a resonator for sensing angular velocity may be used.

FIG. 22 shows a diagram of the feedback circuit of FIG. 21. In this embodiment, the feedback circuit has a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. Now, when angular frequency ω_(i) of the quartz crystal tuning fork resonator 203, a resistance value R_(d) of the drain resistor 207, capacitance values C_(g), C_(d) of the capacitors 205, 206, crystal impedance R_(ei) of the quartz crystal resonator 203, an input voltage V₁, and an output voltage V₂ are taken, a feedback rate β_(i) is defined by β_(i)=|V₂|_(i)/|V₁|_(i), where i shows a vibration order, for example, when i=1 and 2, β₁ and β₂ are a feedback rate for a fundamental mode of vibration and a second overtone mode of vibration of the resonator, respectively.

In addition, load capacitance C_(L) is given by C_(L)=C_(g)C_(d)/(C_(g)+C_(d)), and when C_(g)=C_(d)=C_(gd) and R_(d)>>R_(ei), the feedback rate β_(i) is given by β_(i)=1/(1+kC_(L) ²), where k is expressed by a function of ω_(i) R_(d) and R_(ei). Also, R_(ei) is approximately equal to series resistance R_(i) of the resonator.

Thus, it is easily understood from a relationship of the feedback rate β_(i) and load capacitance C_(L) that the feedback rate of a resonance frequency for the fundamental mode of vibration and the overtone mode vibration becomes large, respectively, according to decrease of load capacitance CL. Therefore, when CL has a small value, an oscillation of the overtone mode occurs very easily, instead of that of the fundamental mode. This is the reason why maximum amplitude of the overtone mode of vibration becomes smaller than that of the fundamental mode of vibration, and the oscillation of the overtone mode satisfies an amplitude condition and a phase condition simultaneously which are the continuous condition of an oscillation in an oscillating circuit.

As it is also one object of the present invention to provide a quartz crystal oscillator having a flexural mode, quartz crystal tuning fork resonator, capable of vibrating in a fundamental mode and having a frequency of high stability (high time accuracy) of an output signal, and having reduced electric current consumption, load capacitance C_(L) is less than 25 pF in this embodiment to reduce electric current consumption. To get much reduced electric current consumption, C_(L) is preferably less than 15 pF because the electric current consumption is proportional to C_(L).

In addition, in order to suppress a second overtone mode of vibration of the resonator and to obtain a quartz crystal oscillator having an output signal of an oscillation frequency of a fundamental mode of vibration of the resonator, the quartz crystal oscillator comprising an amplification circuit and a feedback circuit is constructed so that it satisfies a relationship of α₁/α₂>β₂/β₁>1 and α₁β₁>1, where α₁ and α₂ are, respectively, an amplification rate of the fundamental mode of vibration and the second overtone mode of vibration of the amplification circuit, and β₁ and β₂ are, respectively, a feedback rate of the fundamental mode of vibration and the second overtone mode of vibration of the feedback circuit.

In other words, the quartz crystal oscillator is constructed so that a ratio of the amplification rate α₁ of the fundamental mode of vibration and the amplification rate α₂ of the second overtone mode of vibration of the amplification circuit is greater than that of the feedback rate β₂ of the second overtone mode of vibration and the feedback rate β₁ of the fundamental mode vibration of the feedback circuit, and also a product of the amplification rate α₁ and the feedback rate β₁ of the fundamental mode of vibration is greater than 1. A description of a frequency of high stability in the quartz crystal oscillator will be performed later.

Also, characteristics of the amplifier of the amplification circuit constructing the quartz crystal oscillating circuit of this embodiment can be expressed by negative resistance −RL_(i). For example, when i=1, negative resistance −RL₁ is for a fundamental mode of vibration of the resonator and when i=2, negative resistance −RL₂ is for a second overtone mode of vibration of the resonator. In this embodiment, the quartz crystal oscillating circuit is constructed so that a ratio of an absolute value of negative resistance −RL₁ of the fundamental mode of vibration of the amplification circuit and series resistance R₁ of the fundamental mode of vibration is greater than that of an absolute value of negative resistance −RL₂ of the second overtone mode of vibration of the amplification circuit and series resistance R₂ of the second overtone mode of vibration.

That is, the oscillating circuit is constructed so that it satisfies a relationship of |−RL₁/R₁>|−RL₂|/R₂. By constructing the oscillating circuit like this, an oscillation of the second overtone mode can be suppressed, as a result of which a frequency of oscillation of the fundamental mode of vibration can be output as an output signal because the oscillation of the fundamental mode generates easily in the oscillating circuit. In more detail, an absolute value of the negative resistance −RL₁ is within a range of 60 kΩ to 285 kΩ, and an absolute value of the negative resistance −RL₂ is less than 105 kΩ to suppress the frequency of oscillation of the second overtone mode of the quartz crystal tuning fork resonator in the oscillating circuit and to obtain the frequency of oscillation of the fundamental mode thereof.

In this embodiment, a quartz crystal tuning fork resonator is used, but, instead of the tuning fork resonator, an another contour mode quartz crystal resonator such as a width-extensional mode quartz crystal resonator, a length-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator may be used in a quartz crystal oscillator of the present invention. In this case, a principal vibration of the contour mode quartz crystal resonator is outputted from an oscillating circuit constructing the quartz crystal oscillator through a buffer circuit. In order to suppress a sub-vibration of the contour mode quartz crystal resonator, the quartz crystal oscillator comprising an amplification circuit and a feedback circuit is constructed so that it satisfies a relationship of α_(p)/α_(s)>β_(s)/β_(p) and α_(p)β_(p)>1, where α_(p) and α_(s) are, respectively, an amplification rate of the principal vibration and the sub-vibration of the amplification circuit, and β_(p) and β_(s) are, respectively, a feedback rate of the principal vibration and the sub-vibration of the feedback circuit.

Similar to the quartz crystal tuning fork resonator, for the contour mode quartz crystal resonator, a quartz crystal oscillating circuit of the present invention is constructed so that a ratio of an absolute value of negative resistance −RL_(p) of the principal vibration of the amplification circuit and a series resistance R_(p) of the principal vibration is greater than that of an absolute value of negative resistance −RL_(s) of the sub-vibration of the amplification circuit and a series resistance R_(s) of the sub-vibration. That is, the oscillating circuit is constructed so that it satisfies a relationship of |−RL_(p)|/R_(p)>|−RL_(s)/R_(s). By constructing the oscillating circuit like this, an oscillation of the sub-vibration can be suppressed, as a result of which a frequency of oscillation of the principal vibration can be outputted as an output signal because the oscillation of the principal vibration generates easily in the oscillating circuit. In addition, the principal vibration and the sub-vibration have the same mode of vibration and a different order of vibration each other.

FIG. 23 shows a block diagram of an embodiment of an electronic apparatus of the present invention, and illustrating the diagram of a facsimile apparatus. As shown in FIG. 23, the apparatus generally comprises a modem, a phonetic circuit, a timepiece circuit, a printing portion, a taking portion, an operation portion and a display portion. In this principle, perception and scanning of reflection light of light projected on manuscript in the taking portion are performed by CCD(Charge Coupled Device), in addition, light and shade of the reflection light are transformed into a digital signal, and the signal is modulated by the modem and is sent to a phone line(communication line). Also, in a receiving side, a received signal is demodulated by the modem and is printed on a paper in the print portion by synchronizing the received signal with a signal of a sending side.

In FIG. 23, a quartz crystal resonator is used as a CPU clock of the control portion and the printing portion, as a clock of the phonetic circuit and the modem, and as a time standard of the timepiece. Namely, the resonator constructs a quartz crystal oscillator and an output signal of the oscillator is used. Like this, a plurality of oscillators is used for the electronic apparatus. For example, it is used as a signal to display time at the display portion. In this case, a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode is generally used, and e.g. as the CPU clock, a contour mode quartz crystal resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator is used. To get the facsimile apparatus of this embodiment which operates normally, an accuracy signal of output of the oscillator is required for the facsimile apparatus, which is one of the electronic apparatus of the present invention. Also, a digital display and an analogue display are included in the display of the present invention.

In this embodiment, though the facsimile apparatus is shown as an example of an electronic apparatus of the present invention, the present invention is not limited to this, namely, the present invention includes all electronic apparatus, each of which comprises a quartz crystal oscillator and a display portion at least, for example, cellar phones, telephones, a TV set, cameras, a video set, video cameras, pagers, personal computers, printers, CD players, MD players, electronic musical instruments, car navigators, car electronics, timepieces, IC cards and so forth. In addition, the electronic apparatus may have an another oscillator comprising a piezoelectric resonator for sensing angular velocity made of quartz crystal, ceramics, lithium tantalite and lithium niobate.

Thus, the electronic apparatus of the present invention comprising a display portion and a quartz crystal oscillator at least may operate normally because the quartz crystal oscillator comprises the quartz crystal oscillating circuit with a frequency of high stability, namely, a frequency of high reliability.

Moreover, capacitance ratios r₁ and r₂ of a flexural mode, quartz crystal tuning fork resonator are given by r₁=C₀/C₁ and r₂=C₀/C₂, respectively, where C₀ is shunt capacitance in an electrical equivalent circuit of the resonator, and C₁ and C₂ are, respectively, motional capacitance of a fundamental mode of vibration and a second overtone mode of vibration in the electrical equivalent circuit of the resonator. In addition, the resonator has a quality factor Q₁ for the fundamental mode of vibration and a quality factor Q₂ for the second overtone mode of vibration.

In detail, the tuning fork resonator of this embodiment is provided so that the influence on resonance frequency of the fundamental mode of vibration by the shunt capacitance becomes smaller than that of the second overtone mode of vibration by the shunt capacitance, namely, so that it satisfies a relationship of S₁=r₁/2Q₁ ²<S₂=r₂/2Q₂ ², preferably, S₁<S₂/2, where S₁ and S₂ are called “a stable factor of frequency” of the fundamental mode of vibration and the second overtone mode of vibration. As a result, the tuning fork resonator, capable of vibrating in the fundamental mode and having a frequency of high stability can be provided because the influence on the resonance frequency of the fundamental mode of vibration by the shunt capacitance C₀ is as extremely small as it can be ignored. In this embodiment S₂ has a value greater than 0.13×10⁻⁶ to suppress the second overtone mode of vibration of the resonator.

In addition, as described above, it will be easily understood that the quartz crystal resonator comprising vibrational arms and a base portion, according to the present invention, may have outstanding effects. Similar to this, the quartz crystal unit and the quartz crystal oscillator, according to the present invention, may have also outstanding effects. In addition, the electronic apparatus comprising the quartz crystal oscillator comprising the quartz crystal oscillating circuit having the quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, and having novel shapes, novel electrode construction and excellent electrical characteristics, according to the present invention, may have the outstanding effects. Similar to this, it will be easily understood that the electronic apparatus comprising the quartz crystal oscillator comprising the quartz crystal oscillating circuit having the another contour mode quartz crystal resonator or the thickness shear mode quartz crystal resonator or the resonator for sensing angular velocity, according to the present invention, may have also the outstanding effect. In addition to this, while the present invention has been shown and described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that the changes in shape and electrode construction can be made therein without departing from the spirit and scope of the present invention. 

1.-29. (canceled)
 30. A resonator comprising: a base portion; and a plurality of vibrational arms connected to the base portion, the vibrational arms being vibratable in a flexural mode and including a first vibrational arm and a second vibrational arm, each of the first and second vibrational arms having first and second opposite side surfaces; wherein an electrode is disposed on at least one of the first and second opposite side surfaces of each of the first and second vibrational arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 31. A resonator according to claim 30; wherein each of the first and second vibrational arms has first and second opposite main surfaces; wherein a groove is formed in at least one of the first and second opposite main surfaces of each of the first and second vibrational arms; and wherein at least one mounting arm protrudes from the base portion.
 32. A resonator according to claim 31; wherein each of the first and second vibrational arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein the groove formed in at least one of the first and second opposite main surfaces of each of the first and second vibrational arms is formed in the first width direction of the corresponding one of the first and second vibrational arms.
 33. A resonator according to claim 30; wherein each of the first and second vibrational arms has first and second opposite main surfaces; wherein a groove is formed in at least one of the first and second opposite main surfaces of each of the first and second vibrational arms; wherein each of the first and second vibrational arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein the groove formed in at least one of the first and second opposite main surfaces of each of the first and second vibrational arms is formed in the first width direction of the corresponding one of the first and second vibrational arms.
 34. A resonator according to claim 30; wherein each of the first and second vibrational arms has first and second opposite main surfaces; wherein a groove is formed in each of the first and second opposite main surfaces of each of the first and second vibrational arms; wherein a length of at least one of the grooves formed in the first and second opposite main surfaces of each of the first and second vibrational arms is within a range of 30% to 80% of a length of each of the first and second vibrational arms and is within a range of 0.45 mm to 1.25 mm; and wherein an electrode is disposed in the groove formed in each of the first and second opposite main surfaces of each of the first and second vibrational arms so that the capacitance ratio r₁ of the fundamental mode of vibration of the resonator is less than the capacitance ratio r₂ of the second overtone mode of vibration thereof.
 35. A resonator according to claim 30; wherein the resonator comprises a quartz crystal tuning fork resonator vibratable in a flexural mode of an inverse phase and having a quartz crystal tuning fork base and first and second quartz crystal tuning fork arms connected to the quartz crystal tuning fork base, each of the first and second quartz crystal tuning fork arms having a first main surface and a second main surface opposite the first main surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; and wherein at least one mounting arm protrudes from the quartz crystal tuning fork base and extends in a common direction with the first and second quartz crystal tuning fork arms.
 36. A resonator according to claim 30; wherein the resonator comprises a quartz crystal tuning fork resonator vibratable in a flexural mode of an inverse phase and having a quartz crystal tuning fork base and first and second quartz crystal tuning fork arms connected to the quartz crystal tuning fork base, each of the first and second quartz crystal tuning fork arms having a first main surface and a second main surface opposite the first main surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein the groove formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 37. A resonator according to claim 35; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein at least one of the grooves formed in the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 38. A resonator according to claim 35; wherein an electrode is disposed in the groove formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the quartz crystal tuning fork resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 39. A resonator according to claim 35; wherein the at least one mounting arm is connected to the quartz crystal tuning fork base through a connecting portion and extends in a common direction with the first and second quartz crystal tuning fork arms outside the first and second quartz crystal tuning fork arms.
 40. A resonator according to claim 35; wherein the at least one mounting arm comprises a plurality of mounting arms including a first mounting arm and a second mounting arm protruding from the quartz crystal tuning fork base; wherein each of the first and second mounting arms is connected to the quartz crystal tuning fork base through a connecting portion and extends in a common direction with the first and second quartz crystal tuning fork arms outside the first and second quartz crystal tuning fork arms.
 41. A resonator according to claim 35; wherein the at least one mounting arm is formed between the first and second quartz crystal tuning fork arms.
 42. An oscillator comprising: an oscillating circuit comprising an amplifier, a plurality of capacitors, at least one resistor and a resonator having a base portion and a plurality of vibrational arms connected to the base portion, the vibrational arms being vibratable in a flexural mode and having a first vibrational arm and a second vibrational arm, each of the first and second vibrational arms having first and second opposite side surfaces; wherein an electrode is disposed on at least one of the first and second opposite side surfaces of each of the first and second vibrational arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 43. An oscillator according to claim 42; wherein the resonator comprises a quartz crystal tuning fork resonator vibratable in a flexural mode of an inverse phase and having a quartz crystal tuning fork base and first and second quartz crystal tuning fork arms connected to the quartz crystal tuning fork base, each of the first and second quartz crystal tuning fork arms having a first main surface and a second main surface opposite the first main surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; wherein at least one mounting arm protrudes from the quartz crystal tuning fork base and extends in a common direction with the first and second quartz crystal tuning fork arms; wherein the at least one mounting arm is mounted on a mounting portion of a case for housing the quartz crystal tuning fork resonator and the case is connected to a lid for covering an open end of the case; and wherein an electrode is disposed in the groove formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the quartz crystal tuning fork resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 44. An oscillator according to claim 42; wherein the resonator comprises a quartz crystal tuning fork resonator capable of vibrating in a flexural mode of an inverse phase and having a quartz crystal tuning fork base and first and second quartz crystal tuning fork arms connected to the quartz crystal tuning fork base, each of the first and second quartz crystal tuning fork arms having a first main surface and a second main surface opposite the first main surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein at least one of the grooves formed in the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 45. An oscillator according to claim 43; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein at least one of the grooves formed in the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 46. An electronic apparatus comprising: a display portion; and at least one oscillator comprising an oscillating circuit comprising a resonator having a base portion and a plurality of vibrational arms connected to the base portion, the vibrational arms being vibratable in a flexural mode and including a first vibrational arm and a second vibrational arm, each of the first and second vibrational arms having first and second opposite side surfaces; wherein an electrode is disposed on at least one of the first and second opposite side surfaces of each of the first and second vibrational arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 47. An electronic apparatus according to claim 46; wherein the at least one oscillator comprises a first quartz crystal oscillator comprising a first quartz crystal oscillating circuit having an amplifier, a plurality of capacitors, at least one resistor and a quartz crystal resonator, and a second quartz crystal oscillator comprising a second quartz crystal oscillating circuit having an amplifier, a plurality of capacitors, at least one resistor and a quartz crystal resonator; wherein the quartz crystal resonator of the first quartz crystal oscillating circuit comprises a quartz crystal thickness shear mode resonator vibratable in a thickness shear mode and having a vibrational portion on which a pair of electrodes is disposed; wherein the quartz crystal resonator of the second quartz crystal oscillating circuit comprises a quartz crystal tuning fork resonator vibratable in a flexural mode of an inverse phase and having a quartz crystal tuning fork base and first and second quartz crystal tuning fork arms connected to the quartz crystal tuning fork base; and wherein an electrode is disposed on each of two of side surfaces of each of the first and second quartz crystal tuning fork arms so that a capacitance ratio r₁ of a fundamental mode of vibration of the quartz crystal tuning fork resonator is less than a capacitance ratio r₂ of a second overtone mode of vibration thereof.
 48. An electronic apparatus according to claim 47; wherein each of the first and second quartz crystal tuning fork arms has a first main surface and a second main surface opposite the first main surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein at least one of the grooves formed in the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 49. An electronic apparatus according to claim 47; wherein each of the first and second quartz crystal tuning fork arms has a first main surface and a second main surface opposite the first main surface, and a first side surface and a second side surface opposite the first side surface; wherein a groove is formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms; wherein a first electrode is disposed in the groove formed in each of the first and second main surfaces of each of the first and second quartz crystal tuning fork arms and a second electrode is disposed on each of the first and second side surfaces of each of the first and second quartz crystal tuning fork arms so that the capacitance ratio r₁ of the fundamental mode of vibration of the quartz crystal tuning fork resonator is less than the capacitance ratio r₂ of the second overtone mode of vibration thereof; wherein at least one mounting arm protrudes from the quartz crystal tuning fork base and extends in a common direction with the first and second quartz crystal tuning fork arms; and wherein the at least one mounting arm is mounted on a mounting portion of a case for housing the quartz crystal tuning fork resonator and the case is connected to a lid for covering an open end of the case.
 50. An electronic apparatus according to claim 49; wherein each of the first and second quartz crystal tuning fork arms has a plurality of different widths including a first width and a second width greater than the first width; and wherein at least one of the grooves formed in the first and second main surfaces of each of the first and second quartz crystal tuning fork arms is formed in the first width direction of the corresponding one of the first and second quartz crystal tuning fork arms.
 51. An electronic apparatus according to claim 47; wherein the at least one oscillator comprises a third quartz crystal oscillator comprising a quartz crystal oscillating circuit having an amplifier, a plurality of capacitors, at least one resistor and a quartz crystal resonator vibratable in a flexural mode for sensing angular velocity and having a quartz crystal base portion and at least one quartz crystal vibrational arm connected to the quartz crystal base portion; wherein the at least one quartz crystal vibrational arm has a first main surface and a second main surface opposite the first main surface, and a first side surface and a second side surface opposite the first side surface; wherein at least one groove is formed in each of the first and second main surfaces of the at least one quartz crystal vibrational arm so that a width of the at least one groove formed in each of the first and second main surfaces of the at least one quartz crystal vibrational arm is greater than a distance in the width direction of the at least one groove measured from an outer edge of the at least one groove to an outer edge of the at least one quartz crystal vibrational arm; and wherein a first electrode is disposed in the at least one groove formed in each of the first and second main surfaces of the at least one quartz crystal vibrational arm and a second electrode is disposed on each of the first and second side surfaces of the at least one quartz crystal vibrational arm so that the first electrode has an electrical polarity opposite to an electrical polarity of the second electrode.
 52. An electronic apparatus according to claim 49; wherein the at least one mounting arm comprises a plurality of mounting arms having a first mounting arm and a second mounting arm protruding from the quartz crystal tuning fork base; wherein each of the first and second mounting arms is connected to the quartz crystal tuning fork base through a connecting portion and extends in a common direction with the first and second quartz crystal tuning fork arms outside the first and second quartz crystal tuning fork arms; wherein the case for housing the quartz crystal tuning fork resonator has a first mounting portion and a second mounting portion; wherein the first mounting arm is mounted on the first mounting portion of the case and the second mounting arm is mounted on the second mounting portion of the case, the case being connected to the lid for covering the open end of the case; wherein the display portion comprises a digital display portion; and wherein an output signal of the second quartz crystal oscillating circuit comprising the quartz crystal tuning fork resonator is a clock signal for use in operation of the electronic apparatus to display time information at the digital display portion.
 53. An electronic apparatus according to claim 52; wherein the electronic apparatus comprises one of a phone, a camera and a video camera. 